Answering Questions on Speeds & Waves in Relativistic Space

[kernelpenguin]

Suppose you were in a craft traveling at 0.5c towards an alien planet. That alien planet has been sending out TV broadcasts for the past hundred years. Those broadcasts are moving towards your craft at c.

Will those broadcasts play at twice the speed?

Is this what is meant by 'time dilation at relativistic speeds'? That massless particles from the rest frame are hitting you at twice the rate? Or, rather, the information carried by those particles seems to be twice as fast? Wouldn't that make dilation directional?

And how about if you were moving away from Earth at 0.5c. Would the TV broadcasts play at half the speed?

Furthermore, what is the rate of 'time slowing down in relation to the rest frame' as you approach c? I mean, if 1s = 1s at 0c, then what would 1s be at 0.25c? 0.5c? 0.75c? Is there a function that covers this?

Just a little something that popped into my head the other day when I heard about the SETI signal.

 

[robphy]

Suppose you were in a craft traveling at 0.5c towards an alien planet. That alien planet has been sending out TV broadcasts for the past hundred years. Those broadcasts are moving towards your craft at c.

Will those broadcasts play at twice the speed?

Yes. If you think in terms of pulses, the frequency of received pulses is increased... effectively, "blueshifted" (i.e., higher frequency).

Is this what is meant by 'time dilation at relativistic speeds'? That massless particles from the rest frame are hitting you at twice the rate? Or, rather, the information carried by those particles seems to be twice as fast? Wouldn't that make dilation directional?

Not quite. It's actually the Doppler Effect, although time-dilation does factor into this effect.

And how about if you were moving away from Earth at 0.5c. Would the TV broadcasts play at half the speed?

Yes... analogous to my description above, it's "redshifted" (i.e., lower frequency).

 

[pervect]

What you're calculating is the doppler shift, and it will be sqrt(3), about 1.73, for a velocity of .5c.

It turns out that the doppler shift depends only on the relative velocity between the source and destination.

This is not time dilation, but it can be used to deduce time dilation. For v=.5c, the time dilation factor gamma turns out to be 1/sqrt(3/4) = 2/sqrt(3) = 1.15 (approx).

The fact that the doppler shift for an object moving on a collision path is a constant which depends only on its relative velocity can be used to derive movst of the results of relativity, along with the knowledge that light moves at the speed 'c'. Here is how it might work in your case.

Let the time of impact of the spaceship with the planet be t_pl = t_sh = 0 for both observers.

The planet sends out a radar signal at t_pl =-a before impact. Because the doppler shift factor is constant, we know that it will arrive at some time measured by the ship's clock of t_sh = -a/sqrt(3). The doppler shfit factor not only shifts frequencies upwards by a constant factor, it also (as you note) reduces the time intervals between pulses

The radar signal bounces off the ship at time t_sh = -a/sqrt(3). Because this signal gets doppler shifted by the same amount coming back it arrives back on the planet at t=-a/3.

Now that we know the behavior of light travel times, we can work out the distance, velocity, and detailed observations of a radar observer on the planet.

So, let's let the planet send out two radar pulses, one at t=-2 seconds, and one at t=-1 seconds, and see what it concludes about the arrival of your ship.

The signal send out at t=-2 seconds arrives at t=-2/3 seconds. We can conclude at the midpoint between the emission and arrival of the two signals (which occurs at time t=-4/3 seconds) the ship was a distance of 1/2 the difference between the two signals multiplied by 'c', because we know that light always moves at 'c'. This means that at t=-4/3 seconds, the ship was at a distance of -2/3 light seconds.

The signal sent out at t=-1 seconds arrives at t=-1/3 seconds. This implies that at time t=-2/3 seconds, the ship was a distance of -1/3 of a light second away from the planet.

So in 2/3 of a second, the ship moved 1/3 of a light second towards the planet. This sets the ships velocity at .5c. And at t=0 the ship impacts, so it moves 1/3 of a light second in the last 2/3 of a second as well.

We can deduce the relationship between ship time and planet time by assuming that the ship broadcast it's clock value when it receives the radar pulses. The planet then knows that at t=-4/3 seconds, the ships clock read -2/sqrt(3) seconds, and that at time t=-2/3 seconds, the ships clock read -1/sqrt(3) seconds, and that at time t=0, the ships clock also reads zero.

Thus the planet can deduce that the ships clock must be running slow by a factor of gamma = 2/sqrt(3).

Note that the ship can send out its own radar signals, and come to the same conclusions as the planet - I won't go through the math again, because the calculations are identical.

We can also measure the length of the ship this way (by timing the difference between radar pulses) and come to the conclusion that the length of the ship changes (Lorentz contraction). Note that the ship, upon detailed examination of radar echos of the planet, will come to the conclusion that the planet is Lorentz contracted as well. However, I won't go into the math for this, either, unless there is some interest.